# Questions tagged [complex-dynamics]

This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.

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### Poincaré-Dulac for vector field

In the above calculation of $w(z)$, why would a term containing $\frac{\partial}{\partial z_j}$ appear? Since all of the substitutions in the formula don't have anything containing derivative's.
24 views

### Vector field tangent to a singular holomorphic foliation

Is there an intuitive way to see that the vector field $$\begin{equation} v=\begin{cases} \dot z=e^{\frac{1}{z}}\\ \dot w=e^{\frac{1}{w}} \end{cases} \end{equation}$$ is tangent to a singular ...
16 views

### Complex dynamics: does equicontinuity at a point imply equicontinuity in a neighborhood?

In Alan F. Beardon's book "Iteration of Rational Functions", the author defines the Fatou set $F(R)$ of a rational function $R$ as the maximal open subset of the Riemann Sphere (endowed with the ...
50 views

### Naming bulbs on the Mandelbrot set

Can anyone point me to an article or webiste that explains exactly how the bulbs of the Mandelbrot set are named. I know there are bulbs that have the names "p/q" for every set of co-prime integers. ...
25 views

### Bounding a complex iteration through the logarithmic conjugation

I am going through a paper and the following is stated as obvious. Let $f$ be a holomorphic function and $H = \{x + iy \in \mathbb{C} \;|\; x < log(\epsilon) \}$, where $0 < \epsilon < 1/2$. ...
51 views

### Hartog's Theorem and Entire Functions

I'm interested in multivariable Complex Analysis, and I have two questions: My first question is as follows: after reading about Hartog's Extension Theorem I started wondering about the following ...
13 views

### Vector field tangent to the graphs of solutions of an equation

$\dot{z}=\frac{A(t)}{t^{k+1}}$, $z\in \mathbb{C}^n$, $t\in D_1={|t|<1}, k\in \mathbb{N}(k\geq 1)$, $A(t): D_1 \to \text{End}(\mathbb{C}^n)$ is holomorphic. How to see that the following vector ...
35 views

### Prove that non-attracting periodic orbits of $x \mapsto x^2 + c$ are in the Julia set [closed]

Let $c$ be complex number and let \begin{align} f_c : \mathbb{C} &\to \mathbb{C} \\ x &\mapsto x^2 + c \end{align} be the quadratic map. It is to be shown that if $O$ is a periodic orbit that ...